Stirling 2ª espécie S(n,k) (n≤6)

The Stirling numbers of the second kind S(n,k) calculator for n≤6 quickly computes the number of ways to partition a set of n elements into k non-empty subsets. These numbers are fundamental in combinatorics, especially for counting set partitions. The calculator provides the exact value for any pair (n,k) within the specified range, eliminating manual calculations.

It works based on a precomputed table of Stirling numbers of the second kind for n from 0 to 6. The user selects n (total elements) and k (number of blocks) between 1 and n, and the calculator displays S(n,k). For example, S(4,2)=7, as there are 7 ways to split a set of 4 elements into 2 non-empty subsets. The recursive formula S(n,k)=k*S(n-1,k)+S(n-1,k-1) generates these values, but here a fixed table is used for quick results.

When to use? Ideal for students and professionals solving combinatorial problems, such as assigning tasks to groups, distributing objects into non-empty boxes, or analyzing data structures. Also useful in probability, e.g., calculating probabilities of partition-related events. For n>6, consider using other tools or implementing the recursion.

Caveats: Ensure k does not exceed n, as S(n,k)=0 for k>n. Also, note that Stirling numbers of the second kind count partitions where the order of blocks does not matter. If block order matters, multiply by k! (k factorial) to get the number of assignments. The calculator is limited to n≤6 for simplicity and to avoid large numbers.